We analyze the empirical spectral distribution of random periodic band matrices with correlated entries. The correlation structure we study was first introduced in 2015 by Hochstattler, Kirsch and Warzel, who named their setup "almost uncorrelated" and showed convergence to the semicircle distribution in probability. We strengthen their results which turn out to be also valid almost surely. Moreover, we extend them to band matrices. Sufficient conditions for convergence to the semicircle distribution both in probability and almost surely are provided. In contrast to convergence in probability, almost sure convergence seems to require a minimal growth rate for the bandwidth. Examples that fit our general setup include Curie-Weiss distributed, correlated Gaussian, and as a special case, independent entries.

中文翻译：

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具有相关项的随机带矩阵的几乎肯定的半圆定律

我们分析具有相关条目的随机周期带矩阵的经验谱分布。我们研究的相关结构于 2015 年由 Hochstattler、Kirsch 和 Warzel 首次引入，他们将他们的设置命名为“几乎不相关”，并在概率上显示收敛于半圆分布。我们加强了他们的结果，这些结果几乎肯定也是有效的。此外，我们将它们扩展到波段矩阵。提供了在概率上和几乎肯定上收敛到半圆分布的充分条件。与概率收敛相反，几乎可以肯定的收敛似乎需要最小的带宽增长率。适合我们一般设置的示例包括 Curie-Weiss 分布式、相关高斯，以及作为特殊情况的独立条目。